In the last section, we have succeeded in writing the thermal correlator in terms of correlators on the spatial cylinder. To evaluate (REF ) we need to compute a correlator on the spatial cylinder:
\(\langle \chi \vert \mathcal {O}(0,0)\mathcal {O}(t_{euc}/\tau , t_{euc}/\bar{\tau })\vert \chi \rangle \) .
We shall evaluate it by going to the plane. For simplicity we only work with the holomorphic part, supplementing with the anti-holomorphic part only at the end. Also we assume that we only deal with scalar primaries (ignoring spin for now).
In the limit of large \(c\)  and the ratios \(h_{\mathcal {O}}/c, h_{\chi }/c\)  held fixed, with \(h_{\mathcal {O}}/c \gg h_{\chi }/c\) , we can evaluate the correlator by monodromy method, which can be used for the following configuration of operators on the plane [1]},
\(\langle \chi (0)\chi (x)\mathcal {O}(1)\mathcal {O}(\infty )\rangle = \lim _{z\rightarrow \infty } z^{2h_{\mathcal {O}} } \langle \chi (0)\chi (x)\mathcal {O}(1)\mathcal {O}(z)\rangle \) 
